J. Fluid Mech. (2010), vol. 643, pp. 425–435. c© Cambridge University Press 2010

doi:10.1017/S0022112009992680

425

Horizontal mixing of quasi-uniform straightcompound channel flows

ALESSANDRO STOCCHINO1†AND MAURIZIO BROCCHINI2

1Dipartimento di Ingegneria delle Costruzioni, dell’Ambiente e del Territorio,University of Genova, Via Montallegro 1, 16145 Genova, Italy

2Department of I.S.A.C., Polytechnic University of Marche, Via Brecce Bianche 12, 60131 Ancona, Italy

(Received 28 July 2009; revised 9 October 2009; accepted 9 October 2009)

The generation and evolution of large-scale vortices with vertical axis (macro-vortices)in a straight compound channel under quasi-uniform flow conditions is investigated.We discuss possible similarities and clear differences with free shear layer flows inducedby the meeting of shallow streams of different speeds. An experimental investigationbased on particle image velocimetry (PIV) measurements of free-surface velocitiesforms the basis for an analysis of both the specific features of macro-vortices and of therelated mean flow characteristics. Dynamical properties strongly depend on the ratiorh between the main channel flow depth (h∗

mc) and the floodplain depth (h∗fp), and three

flow classes can be identified. ‘Shallow flows’ (rh > 3) are dominated by strong shearingand large macro-vortices populating the transition region between the main channeland the floodplains. The mean streamwise velocity induced in ‘intermediate flows’(2 � rh � 3) is characterized by a dip in the transition region, while it closely resemblesthat occurring in a rectangular channel in the case of ‘deep flows’ (rh < 2). For boththe latter cases the shear in the transition region decreases and the macro-vortices arealso generated in the wall boundary layer of the floodplains. The typical size of thequasi-two-dimensional macro-vortices, generated at the transition region, is found tobe independent of the streamwise coordinate. This and the non-monotonic behaviourof the mean streamwise velocity suggest that in straight compound channels thetopographic forcing is so dominant that conceptual models interpreting these flows asfree shear layers may largely fail to describe the physics of compound channels flows.

Key words: compound channel, macro-vortices, mixing, shallow flow, shear layer

1. IntroductionA great variety of natural water streams and artificial channels can be classified

as ‘compound channels’ since their cross-stream section can be assumed to becharacterized by a main channel and shallow floodplains. Flows of these streamsare defined as predominantly horizontal since their horizontal dimensions greatlyexceed the vertical one (Jirka 2001; van Prooijen & Uijttewaal 2002). Although inbounded shear flows the three-dimensional turbulent eddy size is, typically, limitedto the shortest dimension (in this case, the water depth), large-scale two-dimensionalcoherent structures with length scales greater than the water depth are observed ina wide range of shallow shear flows (Jirka 2001; Socolofksy & Jirka 2004). Among

† Email address for correspondence: jorma@dicat.unige.it

426 A. Stocchino and M. Brocchini

studies on the evolution of quasi-two-dimensional vortices with vertical axes (macro-vortices) in compound channels important are those of Sellin (1964), Shiono & Knight(1991) and Nezu, Onitsuka & Iketani (1999). The different mechanisms that may leadto the production of vorticity in shallow streams and, ultimately, macro-vortices havebeen recently described by Jirka (2001) and Nikora et al. (2007). Among them, thetopographic forcing that imposes an abrupt gradient in the flow evolution has beenrecognized as the main agent for the generation of macro-vortices. In the same veinSoldini et al. (2004), using the classic shallow-water approach, have illustrated themechanism by which depth discontinuities at the transition between the main channeland the floodplains (either ‘transition’ or ‘transition region’ in the following) are able toproduce potential vorticity and, thus, sustain the generation of macro-vortices. Theseturbulent structures are recognized to be responsible for the transfer of momentumand mass from the main channel to the floodplains. As a consequence, the velocitydecreases in the main channel and increases in the floodplains, possibly resulting in amodified conveyance capacity (van Prooijen, Battjes & Uijttewaal 2005).

In many applications of river engineering and river training works a compoundgeometry is purposely created and, therefore, the estimate of the water discharge,velocity and Reynolds stresses profiles and the prediction of the overbank frontspeed during short duration floods is clearly important (Shiono & Knight 1991; vanProoijen et al. 2005). Moreover, floodplains in natural river are often wetlands withabundant wild life and the knowledge of the mass transport processes is vital topreserve the environmental conditions. In this respect, the large-scale macro-vorticesplay a fundamental role on the transverse exchange of mass (e.g. pollutants, nutrients)between the main channel and the floodplains (van Prooijen & Uijttewaal 2002).

However, until recently very simple conceptual models have been used to representthe mentioned lateral mixing which has been mainly accounted for through anempirical apparent shear stress (e.g. Stephenson & Kolovopoulos 1990). Such aperspective, only adequate for steady-state conditions and based on an integralmodel, is being replaced by an approach for which the major agents of transport ofmass and momentum are the discrete flow structures above defined as macro-vortices.A common theoretical approach used to describe uniform compound channel flowsrelies on the apparent close similarity between the flow at the ‘transition’ and the flowgenerated at the interfaces of two free shallow streams with different velocities. Sucha similarity is based on the evidence that for both these configurations it is possible touse the same sort of stability analysis and also that mean velocity profiles can be, inboth cases, well described by bell-shaped functions (e.g. hyperbolic tangent/secant).

The present contribution focuses on the description of the dynamics undergoneby macro-vortices in a compound channel as part of the overall flow evolution.More specifically, we intend to clarify the validity and ambiguities concerned withthe representation of the flow at the transition in a straight compound channel as afree shear layer flow. Since secondary circulations are shown to be of minor importancein straight uniform compound channel flows (van Prooijen & Uijttewaal 2002; vanProoijen et al. 2005), the experimental analysis which forms the basis of our study isonly concerned with the description of horizontal (at the surface) flow features.

2. Experimental set-up and data analysisPhysical experiments have been carried out at the laboratory of the Department of

Civil, Environmental and Architectural Engineering of the University of Genova usinga flume that is 20 m long, 60 cm wide and whose trapezoidal cross-section is composed

Horizontal mixing of quasi-uniform straight compound channel flows 427

Experiment Name rh (−) Q∗ (l s−1) U ∗m (cm s−1) Re × 104 Fr (−)

Shallow flowsQ008 6.0 3.3 37.4 1.3 0.48Q012 5.7 4.1 37.5 1.4 0.49Q018 4.0 5.0 40.0 1.6 0.50Q027 3.5 6.1 42.7 1.7 0.52

Intermediate flowsQ056 2.7 8.8 46.4 2.1 0.53Q120 2.3 12.9 56.3 2.7 0.60Q180 2.1 15.7 60.7 3.0 0.63

Deep flowsQ287 1.9 19.9 65.0 3.4 0.64Q350 1.8 22.0 68.0 3.6 0.66

Table 1. Hydraulic conditions at x∗ = 7 m.

W*fp = 19 cm W*

fp = 19 cm

h*fpy*x*

5 cm

W*tr = 5 cm W*

tr = 5 cm

W*mc = 20 cm

h*mc

Figure 1. Sketch of the cross-section of the flume.

by a central main channel (W ∗mc = 20 cm), two lateral flat floodplains (W ∗

fp = 19 cm) anda transition region (W ∗

tr = 5 cm) (see figure 1). We use a Cartesian coordinate systemin which x∗ and y∗ are aligned with the streamwise and spanwise direction of the flow,respectively (see also figure 1). In the following, horizontal distances are normalizedby the main channel half-width (W ∗

mc,hw = 10 cm). The compound channel, includingthe lateral vertical walls, is made of sheets of polyvinyl chloride (PVC) 1 cm thick.Three measuring sections have been selected in the longitudinal direction placed at 2,5 and 7 m downstream of the flume inlet. Hydraulic conditions of the experiments,referring to the most downstream section, can be found in table 1, where rh =h∗

mc/h∗fp

is the ratio between the main channel water depth and that of the floodplains, Q∗

is the discharge and U ∗m the peak velocity in the main channel. Re =(R∗U ∗

m)/ν isthe Reynolds number and Fr = U ∗

m/√

gR∗ is the Froude number, where R∗ is thehydraulic radius. The most downstream section, x∗ =7m, is characterized by fullyestablished flow conditions both in terms of bottom boundary layer and transitionshear layer evolution. According to Bousmar et al. (2005) compound channel flowsare fully established at a length L∗ downstream of the inception with L∗/W ∗

fp > 35.In the present case the length L∗ is greater than 6.6 m. Measurements of two-dimensional velocity fields on the free surface have been obtained by means of atwo-dimensional PIV analysis using the IDT software package. An high-speed digitalcamera has been used to record 2000 frames for each acquisition, with acquisitionfrequency between 100 and 200 Hz. The illumination has been provided by threewhite light incandescent lamps of 1000 W each and white plastic particles with a

428 A. Stocchino and M. Brocchini

mean diameter of 300 μm and a specific gravity of about 1.05 were used as tracersfor the PIV. We have preventively treated the particles with ethyl alcohol to avoidclustering on the free surface. This gave an approximate area density of about 10particle cm−2. The area of interest for the flow measurements has dimensions ofabout (0.6 × 0.6)m2 and (1024 × 1024)pixel. Probability density functions (PDFs) ofthe measured displacements have been calculated to verify the occurrence of peak-locking effect. The shape of the PDFs, being almost Gaussian, suggests that themeasured velocity are not affected by this error. The flow depth has been monitoredalong the streamwise direction during each experiment and the differences detectedamong the measuring sections was less than the 1 %.

The resulting two-dimensional velocity vectors have been elaborated to extract themain features of the flow under investigation. A specific analysis to identify and totrack vortical structures has also been carried out. Among the many techniques ofvortex identification, we employed the method based on the evaluation of the swirlingstrength λci , commonly used in turbulence measurements to study coherent structures(Adrian, Christensen & Liu 2000). The swirling strength, defined as the positiveimaginary eigenvalue of the local velocity gradients tensor, is large where there is astrong rotation of the flow, i.e. a vortex. Moreover, to help the visualization of themacro-vortices embedded in the flow we have introduced a Galilean decompositionremoving from the actual flow field a constant longitudinal advection velocity, chosenas the areal-averaged streamwise velocity, as suggested in Adrian et al. (2000). Ifthe constant velocity removed matches the convective velocity of the vortices, thelatter appears as an almost circular pattern of vectors plotting the residual part ofthe velocities. The resulting contour maps of λci have been used to study the maingeometrical features and the evolution of the vortical structures, after having applieda threshold of about 5 % of the maximum value to remove the background noise.In particular, we could infer the position of the core of the vortex, the radius ofthe area-equivalent circle R∗

eq , and other geometrical characteristics of interest, e.g.the eccentricity ε of the vortices and the inclination angle θ with respect to themean flow. The equivalent radius has been normalized (Req) using as a characteristiclength scale the width of the transition region (W ∗

tr ), where the flow experiences thedepth jump. Normalized PDFs of the above features have been obtained. The totalnumber of vortices revealed during a single experiment ranges between 200 and400.

3. Experimental observationsThe main experimental results are concerned with both the mean quantities

(streamwise velocity and Reynolds stress) and the macro-vortices. In particular, thedata presented in the rest of this section regards the most downstream measuringsection, x = 70.

3.1. The mean flow

The two-dimensional velocity vector outputs of the PIV analysis have been elaboratedto extract the main flow features. In particular, we have performed an ensembleaverage on the velocity fields and, then, spatial averages along the streamwisedirection to extract dimensionless mean streamwise velocity (u) profiles and profilesof dimensionless averaged Reynolds stress (u′v′). The velocity scale used to normalizethe above quantities is the velocity at the transition (y = 1). In the following, resultsare analysed with the reference to the classification introduced by Nezu et al. (1999).

Horizontal mixing of quasi-uniform straight compound channel flows 429

y

u

0 0.5 1.51.0 2.0 2.5 3.0

0.8

1.0

1.2

1.4

rh = 6rh = 4rh = 3.5rh = 2.3rh = 2.1rh = 1.9

(a) (b)

(c)

(d)Main channel Floodplain

0 0.5 1.0 1.5 2.0 2.5 3.0

0 0.5 1.0 1.5 2.0 2.5 3.0

0 0.5 1.0 1.5 2.0 2.5 3.0

0.010.020.030.040.050.06

rh = 6rh = 4rh = 3.5

y

0.010.020.030.040.050.06

rh = 1.9

0.010.020.030.040.050.06

rh = 2.3rh = 2.1

u′v′

u′v′

u′v′

Figure 2. Mean streamwise surface velocity (a) and averaged Reynolds stress (b–d ) fordifferent values of rh.

(a) The ‘shallow flows’ (rh > 3) are characterized by monotonic velocity profilesand a strong velocity gradient at the transition between the main channel and thefloodplain (see figure 2a). In fact, the ratio between the peak main channel velocity(y = 0) and the floodplain velocity (y = 1) is of order 1.3–1.5. This leads to a strongshearing in the transition and is associated with intense macro-vortices. At the wallboundary layers only a weak shear occurs and almost no macro-vortices are generated.The spanwise profile of u′v′ (figure 2b) presents a local maximum at y =1 where theflow jump intensifies the generation of turbulence. This feature is found also for thenext class of flows (see figure 2c).

(b) The ‘intermediate flows’ (2<rh < 3) are characterized by non-monotonicvelocity profiles with a dip over the transition (see figure 2a). According to Nezuet al. (1999) this effect is the signature of counter-rotating macro-vortices at thetransition (see also figure 3c). The ratio between the peak main channel velocity andthe floodplain velocity is 1.05–1.15 and, thus, weaker than for the ‘shallow flows’. Thedepth increase leads to a growth of the wall boundary layer thickness and macro-vortices are generated, by wall-adherence-induced shearing, also in the floodplainclose to the walls, as we discuss in the next section.

(c) The ‘deep flows’ (rh < 2) are characterized by a very weak shear in the transitionregion, but there is still influence of the compound channel geometry on the surfacevelocity profiles (see figure 2a). The ratio between the peak main channel velocityand the floodplain velocity is 1.00–1.05. The wall boundary layer increases in sizeand more and stronger macro-vortices are generated over the floodplains. However,the influence of the topography is much weaker and this can also be seen in thedistribution of u′v′ that is flatter except close to the lateral wall (figure 2d ).

3.2. The macro-vortices

In order to better understand the overall flow field and characteristics of macro-vortices, instantaneous measurements of the surface flow velocity are analysed bymeans of vortex identification and vorticity fields, Galilean decomposition of thevelocity field and Power spectral density (PSD) distributions of point-like velocitysignals. Generation of two-dimensional macro-vortices has been observed for all thevalues of rh investigated. The classification of the flow regimes used to characterizethe mean velocity profile can be applied also when the free surface vortical structures

430 A. Stocchino and M. Brocchini

x

–2 –1 0 1 2 –2 –1 0 1 2

–2 –1 0 1 2 –2 –1 0 1 2

70

71

72

73

74

75

70

71

72

73

74

75

70

71

72

73

74

75

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73

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75

λci (s–1)

0.200.180.160.140.120.100.080.060.040.02

λci (s–1)

0.200.180.160.140.120.100.080.060.040.02

0

0

(a)

(c)

(b)

(d)

x

ω (s–1)2.01.61.20.80.40–0.4–0.8–1.2–1.6–2.0

ω (s–1)2.01.61.20.80.40–0.4–0.8–1.2–1.6–2.0

y

1

2

3

4

5

y

Figure 3. Velocity vector fields, after a Galilean decomposition, and swirling strength contourmaps for ‘shallow flows’ (Q008) (a) and ‘intermediate flows’ (Q120) (c); (b) and (d ): thecorresponding vorticity fields.

are considered. In fact, in the case of ‘shallow flows’, inspection of figure 3(a, b)reveals that the transition regions are dominated by large-scale macro-vortices andthe sense of rotation is positive (anticlockwise) for the left shear layer and negative(clockwise) for the right shear layer. The macro-vortices are aligned in the region witha spanwise distance in the range (0.5–1)W ∗

mc,hw from the main-channel axis, whichcorresponds to the region of high turbulence. The macro-vortices remain confined inthe transition region and almost no macro-vortices are present above the floodplains.Their formation can be explained in terms of the vorticity generation mechanismat the depth jumps described in Soldini et al. (2004). Increasing the discharge leadsto the ‘intermediate flows’ for which the surface velocity is characterized by slightlyweaker double shear layers in the transition regions and strong wall shear layersabove the floodplains; this can be seen in the vorticity distribution as well as in thevortex positions in figure 3(c, d ). For example, vortices 1, 2 and 3 are located in thetransition region, vortices 1 and 2 rotate clockwise while vortex 3 rotates anticlockwise.Vortices 4 and 5, both located in the wall shear layers, have negative vorticity and,therefore, rotate clockwise. Finally, for the ‘deep flows’, not shown in the figure,macro-vortices are observed only close to the sidewalls and none is present at the

Horizontal mixing of quasi-uniform straight compound channel flows 431

100 101

–3 –3

f (Hz)

100 101

f (Hz)

10–9

10–8

10–7

10–6

10–5

10–4

10–3

P((

m s

–1)2

/Hz)

10–9

10–8

10–7

10–6

10–5

10–4

10–3

Pvv,mcPvv,slPvv,fp

Pvv,mcPvv,slPvv,fp

–5/3 –5/3

–3

(a) (b)

Figure 4. Power spectral density for cases Q018 and Q120.

transition region. In this case, the difference in the flow depth between the floodplainsand the main channel is no longer an active mechanism to generate vorticity. Forall the experiments, we have never observed vortex breaking and/or vortex merging.The macro-vortices once generated are convected downstream without interactingwith each other. The width of the main channel is such that the vortices at the twotransitional regions are always sufficiently distant. The background shearing of themean flow is not so intense as to produce vortex breaking (Tabeling 2002).

PSD distributions of transverse velocity fluctuations for single point signals areplotted for both a shallow flow case Q018 (see figure 4b) and intermediate flow caseQ120 (see figure 4a). Three spanwise positions were chosen for the analysis of thePSD distributions: the main channel (mc), the shear layer (sl) and the flood plain (fp).Both plots show a −3 regime for frequencies between 1 and 10 Hz, and for frequenciesgreater than 10 Hz the data is not inconsistent with a −5/3 regime. The low-frequencyregime shows that the flow is dominated by large-scale quasi-two-dimensional macro-vortices, both for the ‘shallow flows’ and ‘intermediate flows’ (Jirka 2001; Tabeling2002). The energy scalings found in the present context are consistent with the resultsdiscussed in Nikora et al. (2007) for the subcritical flow conditions. The range of theFroude number investigated does not cover the supercritical regime where a differentscaling has been found in the general case of a free-surface flow (Nikora et al. 2007).

3.3. The main properties of the macro-vortices

In this section, we focus on the main features of the macro-vortices by a statisticalanalysis. It is first fundamental to characterize features of the eddies in dependenceof their generation mechanism which, as above mentioned, is the depth jump inthe transition region for ‘shallow flows’ and the flow shearing at the lateral wallsfor ‘intermediate’ and ‘deep flows’. Macro-vortices of different sizes are generateddepending on the flow depth ratio. To detail such dependencies we have analysedboth the PDFs of the vortex area-equivalent radius (figure 5a–d ) of both transitionregion vortices and wall vortices. For shallow flows (rh > 3), vortices are found to residemainly in the transition region and with typical size in the range Req =0.5−1. For‘intermediate’ and ‘deep flows’ macro-vortices both increase in size and frequency overthe floodplains, whereas the transition regions exhibit a decreasing vortex content.The PDFs of the Req seem to suggests that floodplain vortices tend to be smallerthan those generated at the transition region. A relevant result of the above analysis

432 A. Stocchino and M. Brocchini

0.5 1.0 1.50.5 1.0 1.5

0.5 1.0 1.50.5Req Req

1.0 1.5

ε0.2 0.4 0.6 0.8 1.0

ε0.2 0.4 0.6 0.8 1.0

θ20 40 60

60

80

20 40 80

θ20 40 60 80

20 40 60 80

PD

F(ε

)

0 0

0.1

0.2

0.3

PD

F(R

eq)

0

0.10

0.05

0.15

0.20

PD

F(R

eq)

0

0.10

0.05

0.15

0.20

0

0.10

0.05

0.15

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0

0.10

0.05

0.15

0.20

PD

F(ε

)P

DF

(θ)

PD

F(θ

)

0

0.1

0.2

0.3

0.1

0.2

0.3

0.1

0.2

0.3

0

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0.150.10

0.05

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0.05

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0.05

0

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

0 0

rh = 6 - transition regionrh = 6 - floodplains

rh = 2.3 - transition regionrh = 2.3 - floodplains

rh = 1.8 - transition regionrh = 1.8 - floodplains

rh = 6 - transition regionrh = 6 - floodplains

rh = 4 - transition regionrh = 4 - floodplains

rh = 2.3 - transition regionrh = 2.3 - floodplains

rh = 1.8 - transition regionrh = 1.8 - floodplains

rh = 2.3 - transition regionrh = 2.3 - floodplains

rh = 1.8 - transition regionrh = 1.8 - floodplains

rh = 4 - transition regionrh = 4 - floodplains

rh = 6 - transition regionrh = 6 - floodplains

rh = 4 - transition regionrh = 4 - floodplains

(a)

(e) (f)

(i) (j)

(g) (h)

(k) (l )

(b)

(c) (d)

Figure 5. PDFs of macro-vortex size and shape: equivalent radius Req , vortex eccentricity εand the alignment angle θ for different depth ratios.

Horizontal mixing of quasi-uniform straight compound channel flows 433

is that Req of the transition vortices well compares with the width of the transitionregion itself, as observed numerically by Soldini et al. (2004).

Besides the distribution of the equivalent radius, we have analysed also the shape ofthe vortical structures observed either on the transitional regions or on the floodplains.In particular, we have computed the eccentricity ε, defined as the ratio between themajor and minor axes of the vortex, which is a measure of the vortex symmetry (ε =1represents a perfectly circular vortex). PDFs of ε are reported in figure 5(e–h). In the‘shallow flows’ regime transitional macro-vortices are rather elongated with valuesof ε ≈ 0.55–0.60. For smaller ratios rh, the vortices, which are more concentratedover the floodplains, are characterized by a similar distribution of eccentricity witha primary peak at ε ≈ 0.5 and a secondary maximum at ε ≈ 0.7 In the case of two-dimensional free turbulence, it has been found that elliptical vorticity patches are,in general, unstable. Due to the difference in rotation between the vortex core andthe edges, elliptical vortices are subjected to filamentation and, ultimately, they tendto become circular. This process is known as ‘axisymmetrization’ (see Tabeling 2002,and cited references). In the present case, the latter process is likely to be inhibited bythe topographic forcing and the resulting background shearing, causing the vorticalstructure to remain more elongated.

Finally, we have calculated the PDFs and mean values of the angle θ that expressesthe orientation of the main axis of the vortex with respect to the mean flow (perfectvortex-mean flow alignment is achieved for θ =0) are shown in figure 5(i–l ). For‘shallow flows’ macro-vortices are observed to be oriented to the mean flow with anangle θ ≈ 40◦–50◦. Decreasing rh, the vortices, mainly concentrated along the lateralwalls, are characterized by a bimodal distribution of orientation peaked at θ ≈ 0◦ andθ ≈ 50◦, whereas macro-vortices generated at the transition regions are characterizedby a broader orientation distribution. This suggests that wall vortices are, on average,more aligned with the mean flow than those generated at the transition region.Moreover θ , independent of du/dy over the transition region and weakly decreasingover the floodplain, ranges, in average, between 20◦ and 50◦. Thus, in general, macro-vortices are not parallel to the main flow and those covering the transition region arenot sensitive to the mean shear (a decrease of θ with du/dy is expected for alignmentdue to shearing) and those over the floodplains are weakly aligned by the meanshear. As a consequence, the transition macro-vortices are likely to weakly engulfthe floodplain fluid, more similarly to what happens for wakes than for jets/shearlayers. In fact, the shear at the transition region of a jet is strong enough to alignthe interface vortices parallel to the mean interface; this does not happen for wakes(Mobbs 1968).

4. Discussion and conclusionsA detailed and extensive experimental investigation based on PIV analysis of free-

surface velocities has been made to reveal the fundamental features of the flowcharacterizing a straight compound channel from the inception to fully developedconditions. Specific focus was on elucidating the properties of horizontal flow mixingand pointing out similarities/differences with a true free shear flow. In this respect,a first difference is made evident by the mean flow profile shown in figure 2(a). Formost of the values of the depth ratio rh, a non-monotonic profile characterizes thefully developed flow. Hence, bell-type function, typically used as self-similar solutionsfor free shear layers (Chu, Wu & Khayat 1991; van Prooijen & Uijttewaal 2002), isinadequate to represent the mean flow in a compound channel. Moreover, although

434 A. Stocchino and M. Brocchini

x25 50 75 100 125 150 175 200

0.25

0

0.50

0.75

1.00

1.25

1.75

1.50

2.00

2.25

2.50

PU02 H = 42 mm

PU02 H = 67 mm

Q008

Q012

Q018

Q027

Q056

Q120

Q180

Q287

x20 40 60 80 1000

1

2

3

Q008Q012Q018Q027Q056Q120Q180Q287

Q*(l s–1)

δ

δ

5 10 15 200

1

32

4

6

5Section 2 mSection 5 mSection 7 m

�R

eq�

(a)(b)

(c)

Figure 6. (a) Comparison of the mean equivalent radius 〈Req〉 versus normalized downstreamdistance between the present measurements and the experiments of van Prooijen & Uijttewaal(2002); dimensionless shear layer thickness δ as a function of both the dimensional dischargeQ∗ (b) and the dimensionless downstream distance x (c).

horizontal mixing is governed by the quasi-two-dimensional macro-vortices in bothflows, differences are expected, because the vortical structures of the two flows seem tohave rather different properties. Such a difference is easily appreciated by inspectingfigure 6(a), which illustrates, in suitable dimensionless form, the downstream growthof the mean equivalent radius 〈Req〉, derived from the PDFs shown in the previoussection, for the macro-vortices observed in the present experiments, and thosemeasured in free shear flows by van Prooijen & Uijttewaal (2002). While the formerones have a size almost constant with x, the latter ones grow in size at a considerablerate. This is consistent with the fact that the thickness δ of a free shear layeris expected to grow linearly in the streamwise direction, whereas in the case ofcompound channels δ remains constant, as soon as the flow is fully developed (seefigure 6c). The dependence of δ on the flow discharge is shown in figure 6(b). This is areflection of the fundamental fact that the size of compound channel macro-vorticesscales with the crossflow size of the transition region, which remains constant alongx, (Soldini et al. 2004; Piattella, Brocchini & Mancinelli 2006) whereas shear-layermacro-vortices scale with the shear layer width that increases downstream of theinception (see figure 5 of van Prooijen & Uijttewaal 2002). Moreover, macro-vorticesare also found to reside in different spatial regions in the two flows. In the case of afree shear layer the vortex position, coinciding with that of the shear layer, is seen tobend towards the low-velocity region of the shear layer (van Prooijen & Uijttewaal2002). On the contrary, in a compound channel, not only the lateral walls providean extra agent for vortex generation (mainly confined over the flood plains), but alsotransition-region macro-vortices are found to reside over the transition region only.

In conclusion, the main findings of the present study can be summarized as follows:(a) The global dynamics of a straight compound channel cannot be entirely

explained by a shear layer approach, i.e. the flow largely differs from that obtainedby the meeting of three parallel streams of different velocity. The compoundchannel topography and sidewall boundaries force a non-monotonic velocity profile.Generation and evolution of macro-vortices in compound channel flows is seento strongly depend on the depth ratio rh (Soldini et al. 2004). The observed

Horizontal mixing of quasi-uniform straight compound channel flows 435

macro-vortices have a size almost independent of the streamwise coordinate andthat clearly scales with the crossflow extension of the transition region.

(b) The shape and the orientation of the transitional vortices suggest thatengulfment of floodplain fluid by the main channel flow is more similar to thatof a wake than that of a jet/shear layer.

The results here described will be used as the basis for a theoretical analysis that,in analogy to recent studies of other types of open channel flows (e.g. shallow wakesdiscussed in Negretti et al. 2006), will allow for an analytical description of the meanflow.

Thanks go to Dr A. C. Rummel for contributing to the early stages of this study. Thetwo anonymous referees are thanked for their valuable comments and suggestions.

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